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Properties

Label 63.9.m.b
Level $63$
Weight $9$
Character orbit 63.m
Analytic conductor $25.665$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [63,9,Mod(10,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("63.10");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 63.m (of order \(6\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(25.6648524339\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4}\cdot 3\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} + \beta_{4} - 41 \beta_{2} + \cdots - 41) q^{4}+ \cdots + (6 \beta_{7} + 22 \beta_{6} + \cdots - 818) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} - \beta_{2} + \beta_1) q^{2} + ( - \beta_{6} + \beta_{4} - 41 \beta_{2} + \cdots - 41) q^{4}+ \cdots + (203252 \beta_{7} - 215894 \beta_{6} + \cdots - 13376412) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 4 q^{2} - 164 q^{4} + 840 q^{5} - 140 q^{7} - 6544 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{2} - 164 q^{4} + 840 q^{5} - 140 q^{7} - 6544 q^{8} + 5796 q^{10} - 1784 q^{11} + 9856 q^{14} - 8584 q^{16} + 141456 q^{17} - 257544 q^{19} + 1706256 q^{22} + 348940 q^{23} - 557752 q^{25} + 2913120 q^{26} + 1485092 q^{28} - 4983176 q^{29} - 2376696 q^{31} + 332016 q^{32} - 2491272 q^{35} + 492740 q^{37} + 7088088 q^{38} - 7601832 q^{40} + 4448432 q^{43} + 3678804 q^{44} - 226560 q^{46} - 2704128 q^{47} - 3811024 q^{49} + 15628912 q^{50} + 11135208 q^{52} - 2281460 q^{53} + 6573392 q^{56} + 11442696 q^{58} - 25291140 q^{59} + 59368764 q^{61} - 114153056 q^{64} - 16923396 q^{65} - 107108 q^{67} - 44316972 q^{68} + 19120080 q^{70} + 82809760 q^{71} + 116758404 q^{73} - 72690340 q^{74} + 58244200 q^{77} - 50628092 q^{79} + 93591624 q^{80} - 91061712 q^{82} + 119231208 q^{85} - 39093088 q^{86} + 41392848 q^{88} + 2322516 q^{89} - 151794552 q^{91} - 253819128 q^{92} - 345566088 q^{94} + 172787052 q^{95} - 373659692 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 592x^{6} - 1176x^{5} + 336397x^{4} - 348096x^{3} + 8673408x^{2} + 8271396x + 197880489 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 40665038848 \nu^{7} + 933800395536 \nu^{6} - 23107427488768 \nu^{5} + \cdots - 24\!\cdots\!71 ) / 78\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 199147024 \nu^{7} - 206072832 \nu^{6} - 113162941609 \nu^{5} + 117098450112 \nu^{4} + \cdots - 17\!\cdots\!48 ) / 16\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4986016178489 \nu^{7} + 729234428373612 \nu^{6} + \cdots + 58\!\cdots\!33 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 142450897959062 \nu^{7} - 415028421151641 \nu^{6} + \cdots - 22\!\cdots\!49 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35983628949817 \nu^{7} + 17429303220966 \nu^{6} + \cdots - 87\!\cdots\!11 ) / 21\!\cdots\!10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 206519207416061 \nu^{7} - 209968840103652 \nu^{6} + \cdots + 29\!\cdots\!97 ) / 10\!\cdots\!50 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{4} + \beta_{3} + 296\beta_{2} - \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -3\beta_{7} - 25\beta_{6} + 14\beta_{5} - 544\beta_{3} + 441 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -592\beta_{6} + 592\beta_{4} - 161165\beta_{2} + 1180\beta _1 - 161165 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2364\beta_{7} + 8288\beta_{6} - 16576\beta_{5} - 2364\beta_{4} + 308569\beta_{3} + 435120\beta_{2} - 308569\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -338161\beta_{7} + 321697\beta_{6} + 8232\beta_{5} - 1004365\beta_{3} + 91505156 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3004175\beta_{6} + 4709558\beta_{5} + 1705383\beta_{4} - 346152513\beta_{2} + 175714240\beta _1 - 346152513 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/63\mathbb{Z}\right)^\times\).

\(n\) \(10\) \(29\)
\(\chi(n)\) \(-\beta_{2}\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
11.6264 − 20.1376i
2.77916 − 4.81365i
−2.23583 + 3.87257i
−12.1698 + 21.0787i
11.6264 + 20.1376i
2.77916 + 4.81365i
−2.23583 − 3.87257i
−12.1698 − 21.0787i
−11.1264 − 19.2716i 0 −119.595 + 207.145i 305.183 − 176.198i 0 2236.80 − 872.649i −374.058 0 −6791.21 − 3920.91i
10.2 −2.27916 − 3.94762i 0 117.611 − 203.708i 163.006 − 94.1113i 0 −2345.65 − 512.580i −2239.15 0 −743.031 − 428.989i
10.3 2.73583 + 4.73860i 0 113.030 − 195.774i −586.541 + 338.640i 0 −168.652 + 2395.07i 2637.67 0 −3209.35 − 1852.92i
10.4 12.6698 + 21.9447i 0 −193.046 + 334.366i 538.352 − 310.818i 0 207.497 − 2392.02i −3296.47 0 13641.6 + 7875.98i
19.1 −11.1264 + 19.2716i 0 −119.595 − 207.145i 305.183 + 176.198i 0 2236.80 + 872.649i −374.058 0 −6791.21 + 3920.91i
19.2 −2.27916 + 3.94762i 0 117.611 + 203.708i 163.006 + 94.1113i 0 −2345.65 + 512.580i −2239.15 0 −743.031 + 428.989i
19.3 2.73583 − 4.73860i 0 113.030 + 195.774i −586.541 − 338.640i 0 −168.652 − 2395.07i 2637.67 0 −3209.35 + 1852.92i
19.4 12.6698 − 21.9447i 0 −193.046 − 334.366i 538.352 + 310.818i 0 207.497 + 2392.02i −3296.47 0 13641.6 − 7875.98i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 63.9.m.b 8
3.b odd 2 1 7.9.d.a ✓ 8
7.d odd 6 1 inner 63.9.m.b 8
12.b even 2 1 112.9.s.a 8
21.c even 2 1 49.9.d.c 8
21.g even 6 1 7.9.d.a ✓ 8
21.g even 6 1 49.9.b.a 8
21.h odd 6 1 49.9.b.a 8
21.h odd 6 1 49.9.d.c 8
84.j odd 6 1 112.9.s.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.9.d.a ✓ 8 3.b odd 2 1
7.9.d.a ✓ 8 21.g even 6 1
49.9.b.a 8 21.g even 6 1
49.9.b.a 8 21.h odd 6 1
49.9.d.c 8 21.c even 2 1
49.9.d.c 8 21.h odd 6 1
63.9.m.b 8 1.a even 1 1 trivial
63.9.m.b 8 7.d odd 6 1 inner
112.9.s.a 8 12.b even 2 1
112.9.s.a 8 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 4 T_{2}^{7} + 602 T_{2}^{6} + 1160 T_{2}^{5} + 331700 T_{2}^{4} - 234400 T_{2}^{3} + \cdots + 197796096 \) acting on \(S_{9}^{\mathrm{new}}(63, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 4 T^{7} + \cdots + 197796096 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 77\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 11\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 39\!\cdots\!29 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 21\!\cdots\!69 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 42\!\cdots\!01 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 98\!\cdots\!21 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 15\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 38\!\cdots\!09 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 47\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 20\!\cdots\!89 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 20\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 25\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 40\!\cdots\!49 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 63\!\cdots\!04)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 63\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 13\!\cdots\!89 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 31\!\cdots\!49 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 20\!\cdots\!64 \) Copy content Toggle raw display
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